For a chapter on the Central Limit Theorem; I have a Uniform Distribution problem were a & b are not given. Only the average, standard deviation, and sample size are given. How do you find a & b?
What do a and b stand for?
Find the standard deviation for the given sample data. Round your answer to one more decimal place than is present in the original data.
19, 6, 19, 15, 14, 5, 18, 15, 13
To find the values of a and b in a uniform distribution problem, where only the average, standard deviation, and sample size are given, you can follow these steps:
1. Recall the formula for the average (μ) and standard deviation (σ) of a uniform distribution with parameters a and b:
- Average (μ) = (a + b) / 2
- Standard deviation (σ) = (b - a) / √12
2. Since you are given the average and standard deviation, you can use these equations to set up a system of equations to solve for a and b.
3. Substitute the given values into the equations:
- Average (μ) = (a + b) / 2
- Standard deviation (σ) = (b - a) / √12
4. Rearrange the equations to isolate a and b:
- 2μ = a + b
- √12σ = b - a
5. Solve the system of equations using the given values:
- Substitute the given values for average (μ) and standard deviation (σ):
- 2(μ) = a + b
- √12(σ) = b - a
6. You now have two equations in two variables (a and b). You can solve these equations simultaneously to find the values of a and b.
7. Once you find the values of a and b, you can use them to complete the problem and analyze the uniform distribution.
Remember, the Central Limit Theorem explains the behavior of the mean of a sample as the sample size increases, regardless of the shape of the original distribution. The steps above help you find the values of a and b for a specific uniform distribution problem.